Lecture 7 Vector Space Last Time - Properties of Determinants - Introduction to Eigenvalues - Applications of Determinants - Vectors in Rn Elementary Linear Algebra R. Larsen et al. (5 Edition) TKUEE翁慶昌-NTUEE SCC_10_2007

Lecture 6: Eigenvalue and Vectors Today Vector Spaces and Applications Subspaces of Vector Spaces Spanning Sets and Linear Independence Basis and Dimension Rank of a Matrix and Systems of Linear Equations Reading Assignment: Secs. 4.2-4.6 of Textbook Homework #3 due Next Time Rank of a Matrix and Systems of Linear Equations (Cont.) Coordinates and Change of Basis Applications of Vector Spaces Length and Dot Product in Rn Inner Product Spaces Reading Assignment: Secs 4.7- 5.2

Lecture 6: Elementary Matrices & Determinants Today Vector Spaces Subspaces of Vector Spaces Spanning Sets and Linear Independence Basis and Dimension Rank of a Matrix and Systems of Linear Equations

What Did You Actually Learn about Determinant?

4.2 Vector Spaces Vector spaces: Let V be a set on which two operations (vector addition and scalar multiplication) are defined. If the following axioms are satisfied for every u, v, and w in V and every scalar (real number) c and d, then V is called a vector space. Addition: (1) u+v is in V (2) u+v=v+u (3) u+(v+w)=(u+v)+w (4) V has a zero vector 0 such that for every u in V, u+0=u (5) For every u in V, there is a vector in V denoted by –u such that u+(–u)=0

Scalar multiplication: (6) is in V. (7) (8) (9) (10)

Notes: (1) A vector space consists of four entities: a set of vectors, a set of scalars, and two operations V：nonempty set c：scalar vector addition scalar multiplication is called a vector space (2) zero vector space

Examples of vector spaces: (1) n-tuple space: Rn vector addition scalar multiplication (2) Matrix space: (the set of all m×n matrices with real values) Ex: ：(m = n = 2) vector addition scalar multiplication

(3) n-th degree polynomial space: (the set of all real polynomials of degree n or less) (4) Function space: (the set of all real-valued continuous functions defined on the entire real line.)

Thm 4.4: (Properties of scalar multiplication) Let v be any element of a vector space V, and let c be any scalar. Then the following properties are true.

Notes: To show that a set is not a vector space, you need only find one axiom that is not satisfied. (it is not closed under scalar multiplication) scalar Pf: Ex 6: The set of all integer is not a vector space. integer noninteger Ex 7: The set of all second-degree polynomials is not a vector space. Pf: Let and (it is not closed under vector addition)

Ex 8: V=R2=the set of all ordered pairs of real numbers vector addition: scalar multiplication: Verify V is not a vector space. Sol: the set (together with the two given operations) is not a vector space

Keywords in Section 4.2: vector space：向量空間 n-space：n維空間 matrix space：矩陣空間 polynomial space：多項式空間 function space：函數空間

Lecture 7: Vector Space Today Vector Spaces and Applications Subspaces of Vector Spaces Spanning Sets and Linear Independence Basis and Dimension Rank of a Matrix and Systems of Linear Equations

4.3 Subspaces of Vector Spaces : a vector space : a nonempty subset ：a vector space (under the operations of addition and scalar multiplication defined in V) W is a subspace of V Trivial subspace: Every vector space V has at least two subspaces. (1) Zero vector space {0} is a subspace of V. (2) V is a subspace of V.

Thm 4.5: (Test for a subspace) If W is a nonempty subset of a vector space V, then W is a subspace of V if and only if the following conditions hold. (1) If u and v are in W, then u+v is in W. (2) If u is in W and c is any scalar, then cu is in W.

Ex: Subspace of R2 Ex: Subspace of R3

Ex 2: (A subspace of M2×2) Let W be the set of all 2×2 symmetric matrices. Show that W is a subspace of the vector space M2×2, with the standard operations of matrix addition and scalar multiplication. Sol:

Ex 3: (The set of singular matrices is not a subspace of M2×2) Let W be the set of singular matrices of order 2. Show that W is not a subspace of M2×2 with the standard operations. Sol:

Ex 4: (The set of first-quadrant vectors is not a subspace of R2) Show that , with the standard operations, is not a subspace of R2. Sol: (not closed under scalar multiplication)

Ex 6: (Determining subspaces of R2) Which of the following two subsets is a subspace of R2? (a) The set of points on the line given by x+2y=0. (b) The set of points on the line given by x+2y=1. Sol: (closed under addition) (closed under scalar multiplication)

(b) (Note: the zero vector is not on the line)

Ex 8: (Determining subspaces of R3) Sol:

Thm 4.6: (The intersection of two subspaces is a subspace)

Keywords in Section 4.3: subspace：子空間 trivial subspace：顯然子空間

Lecture 7: Vector Space Today Vector Spaces and Applications Subspaces of Vector Spaces Spanning Sets and Linear Independence Basis and Dimension Rank of a Matrix and Systems of Linear Equations

4.4 Spanning Sets and Linear Independence Linear combination:

Ex 2: (Finding a linear combination) Sol:

(this system has infinitely many solutions)

the span of a set: span (S) If S={v1, v2,…, vk} is a set of vectors in a vector space V, then the span of S is the set of all linear combinations of the vectors in S, a spanning set of a vector space: If every vector in a given vector space can be written as a linear combination of vectors in a given set S, then S is called a spanning set of the vector space.

Notes: Notes:

Ex 5: (A spanning set for R3) Sol:

Thm 4.7: (Span(S) is a subspace of V) If S={v1, v2,…, vk} is a set of vectors in a vector space V, then span (S) is a subspace of V. span (S) is the smallest subspace of V that contains S. (Every other subspace of V that contains S must contain span (S).)

Linear Independent (L.I.) and Linear Dependent (L.D.): : a set of vectors in a vector space V

Notes:

Ex 8: (Testing for linearly independent) Determine whether the following set of vectors in R3 is L.I. or L.D. Sol:

Ex 9: (Testing for linearly independent) Determine whether the following set of vectors in P2 is L.I. or L.D. S = {1+x – 2x2 , 2+5x – x2 , x+x2} v1 v2 v3 Sol: c1v1+c2v2+c3v3 = 0 i.e. c1(1+x – 2x2) + c2(2+5x – x2) + c3(x+x2) = 0+0x+0x2  c1+2c2 = 0 c1+5c2+c3 = 0 –2c1+ c2+c3 = 0   This system has infinitely many solutions. (i.e., This system has nontrivial solutions.)  S is linearly dependent. (Ex: c1=2 , c2= – 1 , c3=3)

Ex 10: (Testing for linearly independent) Determine whether the following set of vectors in 2×2 matrix space is L.I. or L.D. v1 v2 v3 Sol: c1v1+c2v2+c3v3 = 0

 2c1+3c2+ c3 = 0 c1 = 0 2c2+2c3 = 0 c1+ c2 = 0  (This system has only the trivial solution.) c1 = c2 = c3= 0  S is linearly independent. 

Thm 4.8: (A property of linearly dependent sets) A set S = {v1,v2,…,vk}, k2, is linearly independent if and only if at least one of the vectors vj in S can be written as a linear combination of the other vectors in S. Pf: () c1v1+c2v2+…+ckvk = 0  ci  0 for some i

Let vi = d1v1+…+di-1vi-1+di+1vi+1+…+dkvk  d1v1+…+di-1vi-1+di+1vi+1+…+dkvk = 0  c1=d1 , c2=d2 ,…, ci=1 ,…, ck=dk (nontrivial solution) S is linearly dependent Corollary to Theorem 4.8: Two vectors u and v in a vector space V are linearly dependent if and only if one is a scalar multiple of the other.

Keywords in Section 4.4: linear combination：線性組合 spanning set：生成集合 trivial solution：顯然解 linear independent：線性獨立 linear dependent：線性相依

Lecture 7: Vector Space Today Vector Spaces and Applications Subspaces of Vector Spaces Spanning Sets and Linear Independence Basis and Dimension Rank of a Matrix and Systems of Linear Equations

4.5 Basis and Dimension Basis: V：a vector space S ={v1, v2, …, vn}V Linearly Independent Sets Generating Sets Bases S ={v1, v2, …, vn}V S spans V (i.e., span(S) = V ) S is linearly independent  S is called a basis for V Notes: (1) Ø is a basis for {0} (2) the standard basis for R3: {i, j, k} i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1)

(3) the standard basis for Rn : {e1, e2, …, en} e1=(1,0,…,0), e2=(0,1,…,0), en=(0,0,…,1) Ex: R4 {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)} Ex: matrix space: (4) the standard basis for mn matrix space: { Eij | 1im , 1jn } (5) the standard basis for Pn(x): {1, x, x2, …, xn} Ex: P3(x) {1, x, x2, x3}

Thm 4.9: (Uniqueness of basis representation) If is a basis for a vector space V, then every vector in V can be written in one and only one way as a linear combination of vectors in S. Pf: span(S) = V S is linearly independent Let v = c1v1+c2v2+…+cnvn v = b1v1+b2v2+…+bnvn  0 = (c1–b1)v1+(c2 – b2)v2+…+(cn – bn)vn (i.e., uniqueness)  c1= b1 , c2= b2 ,…, cn= bn

Thm 4.10: (Bases and linear dependence) If is a basis for a vector space V, then every set containing more than n vectors in V is linearly dependent. Pf: Let S1 = {u1, u2, …, um} , m > n uiV

Let k1u1+k2u2+…+kmum= 0 (di = ci1k1+ci2k2+…+cimkm) d1v1+d2v2+…+dmvm= 0  di=0 i i.e. Thm 1.1: If the homogeneous system has fewer equations than variables, then it must have infinitely many solution. m > n  k1u1+k2u2+…+kmum = 0 has nontrivial solution  S1 is linearly dependent

Thm 4.11: (Number of vectors in a basis) If a vector space V has one basis with n vectors, then every basis for V has n vectors. (All bases for a finite-dimensional vector space has the same number of vectors.) Pf: S ={v1, v2, …, vn} two bases for a vector space S'={u1, u2, …, un}

Finite dimensional: A vector space V is called finite dimensional, if it has a basis consisting of a finite number of elements. Infinite dimensional: If a vector space V is not finite dimensional, then it is called infinite dimensional. Dimension: The dimension of a finite dimensional vector space V is defined to be the number of vectors in a basis for V. V: a vector space S：a basis for V  dim(V) = #(S) (the number of vectors in S)

S：a generating set  #(S)  n Bases Linearly Independent #(S) > n #(S) = n #(S) < n dim(V) = n Notes: (1) dim({0}) = 0 = #(Ø) (2) dim(V) = n , SV S：a generating set  #(S)  n S：a L.I. set  #(S)  n S：a basis  #(S) = n (3) dim(V) = n , W is a subspace of V  dim(W)  n

Ex: (1) Vector space Rn  basis {e1 , e2 ,  , en}  dim(Rn) = n (2) Vector space Mmn  basis {Eij | 1im , 1jn}  dim(Mmn)=mn (3) Vector space Pn(x)  basis {1, x, x2,  , xn}  dim(Pn(x)) = n+1 (4) Vector space P(x)  basis {1, x, x2, }  dim(P(x)) = 

Ex 9: (Finding the dimension of a subspace) (a) W={(d, c–d, c): c and d are real numbers} (b) W={(2b, b, 0): b is a real number} Sol: (Note: Find a set of L.I. vectors that spans the subspace) (a) (d, c– d, c) = c(0, 1, 1) + d(1, – 1, 0)  S = {(0, 1, 1) , (1, – 1, 0)} (S is L.I. and S spans W)  S is a basis for W  dim(W) = #(S) = 2 (b)  S = {(2, 1, 0)} spans W and S is L.I.  S is a basis for W  dim(W) = #(S) = 1

Ex 11: (Finding the dimension of a subspace) Let W be the subspace of all symmetric matrices in M22. What is the dimension of W? Sol: spans W and S is L.I.  S is a basis for W  dim(W) = #(S) = 3

Thm 4.12: (Basis tests in an n-dimensional space) Let V be a vector space of dimension n. (1) If is a linearly independent set of vectors in V, then S is a basis for V. (2) If spans V, then S is a basis for V. Generating Sets Bases Linearly Independent dim(V) = n #(S) > n #(S) = n #(S) < n

Keywords in Section 4.5: basis：基底 dimension：維度 finite dimension：有限維度 infinite dimension：無限維度

Lecture 7: Vector Space Today Vector Spaces and Applications Subspaces of Vector Spaces Spanning Sets and Linear Independence Basis and Dimension Rank of a Matrix and Systems of Linear Equations

4.6 Rank of a Matrix and Systems of Linear Equations row vectors: Row vectors of A column vectors: Column vectors of A || || || A(1) A(2) A(n)

Let A be an m×n matrix. Row space: The row space of A is the subspace of Rn spanned by the row vectors of A. Column space: The column space of A is the subspace of Rm spanned by the column vectors of A. Null space: The null space of A is the set of all solutions of Ax=0 and it is a subspace of Rn.

Thm 4.13: (Row-equivalent matrices have the same row space) If an mn matrix A is row equivalent to an mn matrix B, then the row space of A is equal to the row space of B. Notes: (1) The row space of a matrix is not changed by elementary row operations. RS(r(A)) = RS(A) r: elementary row operations (2) Elementary row operations can change the column space.

Thm 4.14: (Basis for the row space of a matrix) If a matrix A is row equivalent to a matrix B in row-echelon form, then the nonzero row vectors of B form a basis for the row space of A.

Find a basis of row space of A = Ex 2: ( Finding a basis for a row space) Find a basis of row space of A = Sol: A= B =

a basis for RS(A) = {the nonzero row vectors of B} (Thm 4.14) = {w1, w2, w3} = {(1, 3, 1, 3) , (0, 1, 1, 0) ,(0, 0, 0, 1)} Notes:

Ex 3: (Finding a basis for a subspace) Find a basis for the subspace of R3 spanned by Sol: A = G.E. a basis for span({v1, v2, v3}) = a basis for RS(A) = {the nonzero row vectors of B} (Thm 4.14) = {w1, w2} = {(1, –2, – 5) , (0, 1, 3)}

Ex 4: (Finding a basis for the column space of a matrix) Find a basis for the column space of the matrix A given in Ex 2. Sol. 1:

CS(A)=RS(AT) a basis for CS(A) = a basis for RS(AT) = {the nonzero vectors of B} = {w1, w2, w3} (a basis for the column space of A) Note: This basis is not a subset of {c1, c2, c3, c4}.

Sol. 2: Leading 1 => {v1, v2, v4} is a basis for CS(B) {c1, c2, c4} is a basis for CS(A) Notes: (1) This basis is a subset of {c1, c2, c3, c4}. (2) v3 = –2v1+ v2, thus c3 = – 2c1+ c2 .

Thm 4.16: (Solutions of a homogeneous system) If A is an mn matrix, then the set of all solutions of the homogeneous system of linear equations Ax = 0 is a subspace of Rn called the nullspace of A. Pf: Notes: The nullspace of A is also called the solution space of the homogeneous system Ax = 0.

Ex 6: (Finding the solution space of a homogeneous system) Find the nullspace of the matrix A. Sol: The nullspace of A is the solution space of Ax = 0. x1 = –2s – 3t, x2 = s, x3 = –t, x4 = t

Thm 4.15: (Row and column space have equal dimensions) If A is an mn matrix, then the row space and the column space of A have the same dimension. dim(RS(A)) = dim(CS(A)) Rank: The dimension of the row (or column) space of a matrix A is called the rank of A. rank(A) = dim(RS(A)) = dim(CS(A))

Nullity: The dimension of the nullspace of A is called the nullity of A. nullity(A) = dim(NS(A)) Notes: rank(AT) = rank(A) Pf: rank(AT) = dim(RS(AT)) = dim(CS(A)) = rank(A)

Thm 4.17: (Dimension of the solution space) If A is an mn matrix of rank r, then the dimension of the solution space of Ax = 0 is n – r. That is n = rank(A) + nullity(A) Notes: (1) rank(A): The number of leading variables in the solution of Ax=0. (The number of nonzero rows in the row-echelon form of A) (2) nullity (A): The number of free variables in the solution of Ax = 0.

Notes: If A is an mn matrix and rank(A) = r, then Fundamental Space Dimension RS(A)=CS(AT) r CS(A)=RS(AT) NS(A) n – r NS(AT) m – r

Ex 7: (Rank and nullity of a matrix) Let the column vectors of the matrix A be denoted by a1, a2, a3, a4, and a5. a1 a2 a3 a4 a5 (a) Find the rank and nullity of A. (b) Find a subset of the column vectors of A that forms a basis for the column space of A . (c) If possible, write the third column of A as a linear combination of the first two columns.

Sol: Let B be the reduced row-echelon form of A. a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 (a) rank(A) = 3 (the number of nonzero rows in B)

(b) Leading 1 (c)

Thm 4.18: (Solutions of a nonhomogeneous linear system) If xp is a particular solution of the nonhomogeneous system Ax = b, then every solution of this system can be written in the form x = xp + xh , wher xh is a solution of the corresponding homogeneous system Ax = 0. Pf: Let x be any solution of Ax = b. is a solution of Ax = 0

Ex 8: (Finding the solution set of a nonhomogeneous system) Find the set of all solution vectors of the system of linear equations. Sol: s t

i.e. is a particular solution vector of Ax=b. xh = su1 + tu2 is a solution of Ax = 0

Thm 4.19: (Solution of a system of linear equations) The system of linear equations Ax = b is consistent if and only if b is in the column space of A. Pf: Let be the coefficient matrix, the column matrix of unknowns, and the right-hand side, respectively, of the system Ax = b.

Then Hence, Ax = b is consistent if and only if b is a linear combination of the columns of A. That is, the system is consistent if and only if b is in the subspace of Rm spanned by the columns of A.

Notes: If rank([A|b])=rank(A) Then the system Ax=b is consistent. Ex 9: (Consistency of a system of linear equations) Sol:

c1 c2 c3 b w1 w2 w3 v (b is in the column space of A) The system of linear equations is consistent. Check:

Summary of equivalent conditions for square matrices: If A is an n×n matrix, then the following conditions are equivalent. (1) A is invertible (2) Ax = b has a unique solution for any n×1 matrix b. (3) Ax = 0 has only the trivial solution (4) A is row-equivalent to In (5) (6) rank(A) = n (7) The n row vectors of A are linearly independent. (8) The n column vectors of A are linearly independent.

Keywords in Section 4.6: row space : 列空間 column space : 行空間 null space: 零空間 solution space : 解空間 rank: 秩 nullity : 核次數

Lecture 6: Eigenvalue and Vectors Today Vector Spaces and Applications Subspaces of Vector Spaces Spanning Sets and Linear Independence Basis and Dimension Rank of a Matrix and Systems of Linear Equations Reading Assignment: Secs. 4.2-4.6 of Textbook Homework #3 due Next Time Rank of a Matrix and Systems of Linear Equations (Cont.) Coordinates and Change of Basis Applications of Vector Spaces Length and Dot Product in Rn Inner Product Spaces Reading Assignment: Secs 4.7- 5.2